I need help writing a critical appreciation of the poem "My Last Duchess".

A critical appreciation focuses on the literary qualities of the poem rather than simply being a summary.

Your introduction might mention that the poem was written by Robert Browning and first published in 1842. It is an historical poem, being set in Renaissance Italy rather than England of the Victorian era.

Next, you might discuss meter. The poem consists of 28 heroic couplets meaning that the lines are written in iambic pentameter and have the rhyme scheme AABBCC, etc. It has no stanza breaks. Browning's frequent use of enjambment, in which syntactical breaks do not coincide with line breaks but rather sentences run unbroken from one line to the next, make the poem flow in almost a pattern of ordinary speech, unlike, say, the work of Alexander Pope in which the lines are end-stopped and rhymes emphasized often for humorous effect. 

The genre of the poem is a dramatic monologue, in which we overhear the speaker talking to a messenger. As the speaker talks, he reveals elements of his character which gradually undermine his credibility, revealing himself as selfish, cruel, jealous, paranoid, and perhaps a murderer. 

A theme this poem has in common with many others by Browning is that of the "collector", a character who is obsessed with collecting, owning, and possessing, and who treats people as possessions.